Logarithm of Infinite Product of Real Numbers

Theorem

Let $(a_n)$ be a sequence of strictly positive real numbers.

Convergence

The following are equivalent:

The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ converges to $a \in \R_{\ne 0}$.

The series $\displaystyle \sum_{n \mathop = 1}^\infty \ln a_n$ converges to $\ln a$.

Divergence to zero

The following are equivalent:

• The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $0$.

• The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $-\infty$.

Divergence to infinity

The following are equivalent::

$(1): \quad$ The infinite product $\displaystyle \prod_{n \mathop = 1}^\infty a_n$ diverges to $+\infty$.

$(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log a_n$ diverges to $+\infty$.

Also see

• Logarithm of Infinite Product of Complex Numbers